Babylonian Word Problems

In terms of practicality, which to me means something is useful and in terms of math has real world context, the Babylonians word problems appeared to be practical in the sense that they were related to real life tasks and topics. However it seems as though they were not practical at all given that the dimensions and calculations given in the word problems were highly unlikely in the real world, particularly during that time. I don’t think the idea of practicality relies on our familiarity with algebra, as to make something such as a word problem practical, real world values and measurements should be used, which I believe is possible to carry out regardless of the knowledge of algebra available. 

When it comes to generality and the ability for the formula or problem to be applicable across several situations, it seems as though the properties of Babylonian mathematics were not very generalizable. This is given that the goal of Babylonian mathematics was to  train students in particular methods, not to generalize their mathematics to solve new problems.  I do, however, think the idea of practicality relies on our familiarity with algebra, as using it may make it easier to generalize previous mathematics to other problems. 

In terms of abstraction, I believe that Babylonians could carry out this idea similar to us given the lack of algebra familiarity, as abstraction is based on constructing or extracting a diagram or pattern from the mathematics. Given the Babylonian’s extensive use of geometry and diagrams, I don’t think the knowledge of contemporary algebra would have affected their abstraction techniques.

When it comes to the idea of pure vs applied mathematics, although Babylonian mathematics may have been ‘practical’ in the sense that they are applied in form, they are still pure in substance as the reading addresses. Thus, although Babylonian’s were only applying the mathematical problems, the problems that they created had pure matter. Potentially with more knowledge surrounding the concept of algebra they could have had more pure based problems. 

Babylonian 'Algebra'

Reading this piece it was interesting to see how math was done in a time before the development of algebra and algebraic notation. It is amazing how they solved similar problems to us without using algebra, something I cannot even imagine doing. However, I would imagine that is because I do not know any different and learned how to solve mathematical questions using algebra and algebraic expressions. In a time before the development of algebra and algebraic notation, the general mathematical principles could be stated using operations that could then be carried out using the different types of tables the Babylonian’s had developed. Although Babylonian’s may not have had algebraic expressions, I believe they could have likely generalized mathematical principles using the different types of tables they had created. 

This is a tricky question! I do not think mathematics is all about generalization and abstraction. There are very concrete aspects of mathematics, however those concrete concepts and facts can then be generalized and applied abstractly to different contexts. Babylonian’s seemed to be solving problems for a specific purpose, therefore I do not think generalizations and abstractions were a particularly prevalent part of the mathematics they carried out. However, I would imagine that while Babylonian’s were using their more concrete ideas and concepts to solve problems, generalizations and abstractions to the problems and concepts being analyzed were occurring, as a way to to adapt their previously used methods to new problems that presented themselves. 

Although it is hard to imagine stating mathematical relationships without algebra, I think areas such as geometry and graph theory would be reasonable areas to consider without the use of algebra. This is because these areas require more drawing and visualizing, making it less dependent on algebra should it not be available. 

Babylonian Style Base 60 Multiplication table for 45

 

Col I	Col II
2, 30	18
4	11, 15
6	7, 30
10	4, 30
16	3, 45

Base 60

I think 60 may have been thought of as a convenient number to use for a base number notational system because it has quite a few factors. Thus it may have been thought of as a useful number as it may have made doing operations such as division and multiplication easier, as there are many factors to choose from. 

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

Where as, the number 10 has much fewer factors. 

Factors of 10: 1, 2, 5, 10

Because of the smaller number of factors in 10, it may have seemed like a less useful number to use for a base. Given that our learning was done through a base 10 system, it seems like a more convenient and useful base to me as I find the factors of 10 to be easy to use in mathematical operations. 

It is interesting to look at the significance of the number 60 now in our daily lives. Given that there are 60 seconds in a minute and 60 minutes in an hour, we rely heavily on the number 60 in our everyday lives, especially in terms of time. 

Given the research that I have done, it is believed that 60 was chosen as the base, given it is the smallest number divisible by 1, 2, 3, 4, 5 and 6, making it easier to express fractions. By using 60 as the base it also allowed for the use of a formula to conduct multiplication, only having to know the squares, instead of using times tables. Today the base 60 system is still used to measure geographic coordinates and angles. As well, our system of time comes from the base 60 Babylonian numeration system. 

Lombardi, M. A. (March 5, 2007). Why is a minute divided into 60 seconds, an hour into 60 minutes, yet there are only 24 hours in a day. Scientific American. https://www.scientificamerican.com/article/experts-time-division-days-hours-minutes/#:~:text=The%20Babylonians%20made%20astronomical%20calculations,the%20first%20six%20counting%20numbers 

Gill, N.S. (July 3, 2009). Babylonian Mathematics and the Base 60 System. Thought Co. https://www.thoughtco.com/why-we-still-use-babylonian-mathematics-116679 

The Crest of the Peacock

While reading this piece, one thing that surprised me was the lack of acknowledgement of Arab contribution to not only European Mathematics but also to philosophy, natural sciences and medicine. I found this surprising based off of the fact that all of the math history I had learned prior to this class had only discussed European and Greek mathematicians or contributions to mathematics. It made me realize that even now, from what I have experienced, Arab contributions are not being talked about enough, despite being very important in the advancement of mathematics. 

I also found the mention of how cultural and geographical barriers had very little hindrance on the cross-transmission of mathematical knowledge. I found this surprising as I would imagine the language, cultural and geographical barriers, especially during those ages would have made it difficult for transmission of knowledge to occur, particularly in math. Learning this gives me even more reason to be excited about math as I am reminded that mathematicians will go to great lengths to help make advancements in this field. 

Finally I found the mention of the possibility that around two hundred and fifty years prior to European mathematicians deriving the infinite series for pi, Madhava of Kerala may have done it. I found this surprising as I have only ever heard mention of Europe's connection to pi, not that of medieval India. This finding makes me realize how important it is to teach math history, and how much I want to include it in my math teaching. Even if it is just small bits of information occasionally to provide correct historical information or present all possibilities if the history of the math is not entirely clear.

Teaching Math History

I believe math history should be incorporated into my own math teaching. I think it is important to teach math history as it can provide the opportunity for students to feel more connected to the math they are learning. Students can gain a greater understanding of how different fields of mathematics were developed, as well as learning about influential mathematicians. I always enjoyed when my teachers would incorporate information about mathematicians into lessons, whether that was stories about their lives, how they got involved in math or how they came to discover and add to different fields of mathematics. I believe it can help engage students in the material, especially when discussing mathematicians as they often had very unique and interesting lives. I think a way I could include math history in my teaching is at the beginning of some topics, starting with information or story about a mathematician who was an integral part of advancing that field of mathematics. As well, if the curriculum allows for it, having the student pick a field of early mathematics that they are interested in and completing some form of project on it.

In the arguments against the incorporation of math history, I disagreed with the idea that because there is no clear way of assessing the history learned in a math class, there is no point in teaching it as students will not pay attention. I believe including math history even with no testable component, is still a great way to provide context and allow students to connect while gaining a greater interest in the material. However, I connected with the argument in favour of teaching math history, which suggests including historical content can in turn motivate students to take charge of their learning, by asking questions about the new material and conducting research on a topic or fact they found interesting. I felt this connected to the type learning that students are now being encouraged to participate in, to follow their interests through self directed learning. Prior to reading this piece, I hadn't considered a lack of availability to accessible and understandable math history resources, as a reason why some teachers may choose not to incorporate it into their courses. As well, I was introduced to new ways math history could be included in the classroom.